862
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 3, SEPTEMBER 2011
Analytical Method for Magnetic Field Calculation in
a Low-Speed Permanent-Magnet Harmonic Machine
Linni Jian, Member, IEEE, Guoqing Xu, Chunting Chris Mi, Senior Member, IEEE,
K. T. Chau, Senior Member, IEEE, and C. C.Chan, Fellow, IEEE
Abstract—Magnetic-gearing effect has become increasingly attractive when designing direct-drive low-speed permanent-magnet
machines. The machines derived from the magnetic-gearing effect can be termed as harmonic machines. Unlike the conventional
types, harmonic machines rely on the field harmonics to achieve
energy conversion and transmission. The detailed knowledge of
the field distributions in the air gap is vitally important for predicting and optimizing its performance. In this paper, we present
an analytical approach to calculate the magnetic field distribution
in a low-speed permanent-magnet harmonic machine. A series-slot
model which is composed of a group of partial differential equations concerning the scalar magnetic potential is built up. Then,
the field solutions are obtained by using the method of separating
variables and analyzing the field boundary conditions. Finally, the
flux densities are derived from the scalar magnetic potentials. All
the results agree well with those obtained from the finite element
method.
Index Terms—Analytical calculation, direct drive, harmonic machine, low speed, magnetic gearing, permanent magnet.
I. INTRODUCTION
OW-SPEED permanent-magnet (PM) machines have attracted increasing attention in several occasions, such as
wind power generation, electric vehicles, electric vessels, home
appliances, etc., since they can offer the so-called direct-drive
operation which is the benefit for avoiding the nuisances caused
by mechanical gearboxes. In order to result in relatively lowrated speed, multipolar PMs are usually adopted. In [1], a 48pole 144-slot outer-rotor surface-mounted PM generator was
proposed for directly coupled wind turbines. A comparative
study showed that it possesses several attractive merits, such
as compact structure, light weight, and high efficiency [2].
In [3], a 24-pole 72-slot axial-flux PM motor was presented
L
Manuscript received October 7, 2010; revised December 3, 2010; accepted
March 30, 2011. Date of publication May 10, 2011; date of current version
August 19, 2011. Paper no. TEC-00392-2010
L. Jian is with the Shenzhen Institutes of Advanced Technology, Chinese
Academy of Sciences, Shenzhen 518055, China (e-mail: ln.jian@siat.ac.cn).
G. Xu is with the The Chinese University of Hong Kong, Shatin, Hong Kong,
and also with the Department of Electrical Engineering, Tongji University,
Shanghai, China (e-mail: gqxu@mae.cuhk.edu.hk).
C. C. Mi is with the Department of Electrical and Computer Engineering, University of Michigan-Dearborn, Dearborn, MI 48128 USA (e-mail:
chrismi@umich.edu).
K. T. Chau and C. C. Chan are with Electrical and Electronic Engineering
Department, The University of Hong Kong, Pokfulam, Hong Kong (e-mail:
ktchau@eee.hku.hk; ccchan@eee.hku.hk).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEC.2011.2140373
for a machine-room-less elevator system. In order to improve
torque density, a 30-pole 60-slot double-rotor axial-flux PM
motor was proposed for electric vehicles [4]. In these machines,
distributed windings were employed, which result in enormous
slot numbers. This will definitely cause problems, such as deep
saturation [5], high temperature rise [6], and so on. By adopting
fractional slot and concentrated windings [7]–[9], these shortcomings can be alleviated. Moreover, it can reduce the volume of the coil end, decrease the copper loss, and improve
the flux-weakening operation ability [10]. Although there are
differences on the winding structure and the field path, all the
aforementioned machines rely on the fundamental component
of the electromagnetic field to realize electromechanical energy
conversion.
Recently, the magnetic-gearing effect has become more and
more attractive when designing low-speed PM machines [11],
[12]. Coaxial magnetic gears [13], [14] rely on the field modulation arising from the noneven magnetic field path to excite abundant asynchronous harmonic components. With the interaction
of specific harmonics [15], both torque transmission and speed
variation can be achieved. In [16], a magnetic-geared machine
was proposed for wind power generation, in which a PM highspeed brushless machine was artfully integrated into a coaxial
magnetic gear to realize low-speed operation. Moreover, it can
offer very large torque density due to the torque amplification
aroused by the magnetic gear. Then, a flux-modulated machine
was proposed [17], in which, the high-speed rotor adopted in
the magnetic-geared machine was removed in order to simplify
the overall structure. Thus, the asynchronous field harmonics
are directly involved into the electromechanical energy conversion process. Although the field harmonics are usually weaker
than the field fundamental component, quantitative comparison showed that this machine can provide higher torque density
than its conventional counterparts [18]. In [19], the field modulation is achieved by directly equipping flux-modulation poles on
the stator. Thus, only one air gap is needed. Calculation results
showed that it can further improve the torque density. Unlike the
conventional machines, these newly emerging types rely on the
field harmonics to achieve energy conversion and transmission.
Thus, they can be termed as harmonic machines in general.
As is well known that the detailed knowledge of the magnetic field distribution in air gap is vitally important for design
and optimization of PM machines, especially for low-speed
PM harmonic machines. Although numerical tools, such as finite element method (FEM), are able to offer precisely field
prediction, they can provide neither closed-form solution nor
physical insight. In recent years, much progress has been made
0885-8969/$26.00 © 2011 IEEE
JIAN et al.: ANALYTICAL METHOD FOR MAGNETIC FIELD CALCULATION IN A LOW-SPEED PERMANENT-MAGNET HARMONIC MACHINE
863
harmonics, and the PPN of these harmonics can be expressed as
PPNi,j,k = |(2i − 1) pk + jNs |
(1)
where i = 1, 2, 3, ..., ∞, j = 0, ±1, ±2, ±3, ..., ±∞, k = 1, 2,
and k = 1 implies that the harmonic component is excited by
the outer rotor PMs, while k = 2 means that it is excited by the
inner rotor PMs.
For each harmonic, the corresponding rotational speed can be
given by
ωi,j,k =
(2i − 1) pk
ωk
(2i − 1) pk + jNs
(2)
where ω 1 and ω 2 are the rotational speeds of the outer rotor and
the inner rotor, respectively.
Stable torque transmission can be achieved between the two
rotors, as long as it satisfies
Ns = p1 + p 2
(3)
and the rotational speeds of the two rotors are governed by
Fig. 1. Magnetic-gearing effect. (a) Coaxial magnetic gear. (b) Magneticgeared machine. (c) Flux-modulated machine. (d) Harmonic machine.
in analytical modeling of PM machines. Generally, there are
two sorts of analytical methods for calculating magnetic field
distribution in PM machines: one is based on conformal transformation [20]–[22], while the other is based on Fourier series decomposition [23]–[27]. The purpose of this paper is to
propose an analytical method for magnetic field calculation in
low-speed PM harmonic machines. Fourier series decomposition will be engaged to obtain the solutions of Laplace/Poisson
equations that represent the field behavior in each calculation region. Moreover, a series-slot model will be employed to improve
the calculation accuracy. The validity of the proposed analytical
approach will be verified by comparing the calculation results
with those obtained from the FEM.
II. PM HARMONIC MACHINES
PM harmonic machines are derived from the magneticgearing effect. Fig. 1(a) shows the topology of a coaxial magnetic gear (CMG). It consists of two rotational parts: the inner
rotor (high-speed rotor) and the outer rotor (low-speed rotor),
and one stationary part: the modulating ring. PMs are employed
on the two rotors. p1 and p2 denote the pole-pair numbers (PPN)
of PMs on the outer rotor and the inner rotor, respectively. The
modulating ring is composed of several ferromagnetic segments
that are symmetrically deployed in the space between the two
rotors. In order to enforce the structural strength for high-torque
transmission, epoxy is filled in the slot area.
The modulating ring takes charge of modulating the magnetic
fields in the two air gaps beside it. Denoted by Ns the number
of ferromagnetic segments, the magnetic field in the both air
gaps built up by the PMs on the two rotors consists of abundant
ω2 = −
p1
ω1 = −Gr ω1
p2
(4)
where Gr is the so-called gear ratio. The minus sign indicates
that the two rotors rotate in opposite directions.
Since all the PMs are involved into the torque transmission,
the CMG can provide as high-torque density as its mechanical
counterparts. Moreover, the noncontact torque transmission can
avoid the nuisances existing in the mechanical gearboxes, such
as friction loss, audible noise, mechanical vibration, and need
of regular lubrication and maintenance. Fig. 1(b) shows the
topology of the magnetic-geared machine. A PM high-speed
brushless machine is artfully placed in the inner bore of the
CMG, and shares its rotor with the CMG (inner rotor of the
CMG). Although it can provide both very high torque density
and low-speed operation, its mechanical structure is too complex: it consists of two rotational parts, two stationary parts,
and three air gaps. Fig. 1(c) shows the topology of the fluxmodulated machine. It can be seen that the inner rotor existing in Fig. 1(b) is removed, and the harmonics excited by the
modulating ring directly interact with the armature windings
to achieve electromechanical energy conversion. This machine
can also provide low-speed operation, and it can offer higher
torque density than the conventional PM machines. However,
it is still composed of one rotational part, two stationary parts,
and two air gaps. In order to further simplify the mechanical
structure, as illustrated in Fig. 1(d), the ferromagnetic modulating poles (FMPs) are directly installed on the stator of the
machine; thus, only one air gap is needed. Moreover, centralized armature windings are employed to improve air ventilation
and heat dissipation, and reduce copper loss and manufacturing cost. Similar to the flux-modulated machine, this harmonic
machine also replies on the field harmonics to achieve energy
conversion and transmission. In what follows, the magnetic field
distribution in it will be theoretically analyzed.
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Fig. 2.
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 3, SEPTEMBER 2011
Analytical model in a pseudopolar coordinate.
III. ANALYTICAL MODEL
In the harmonic machine shown in Fig. 1(d), the PPN of PMs
on the outer rotor equals 24 (p1 = 21), and the three-phase armature windings has six poles (p2 = 3). Moreover, the number of
FMPs is 24. Thus, the gear ratio of 7:1 is resulted. The series-slot
model within a pole-pair pitch can be represented in the pseudopolar coordinate as shown in Fig. 2. The widths of the slots,
openings, and FMPs are denoted by g, f, and e, respectively. r1 ,
r2 , r3 , and r4 are the inside radius of the openings, outside radius
of the FMPs, inside radius of the PMs, and outside radius of the
PMs, respectively. The θ-axis coordinates of the central lines
of the kth FMP Tk , ith opening Oi , and jth slot Sj are denoted
by αk , β i , and γ j , respectively. ω t denotes the rotational speed
of the outer rotor. Moreover, for simplicity, several assumptions
should be made.
1) The permeabilities of the iron yokes are infinite.
2) The PMs have the same permeability with the airspace.
3) The depth of the slots is infinite.
The calculation region can be classified into four parts: PMs
(Region I), air gap (Region II), openings between the FMPs
(Region III), and slots (Region IV). In various regions, the flux
density B and field intensity H are expressed as
in Region I:
B = μ0 μr H + μ0 M
in Regions II, III, IV:
(5)
B = μ0 H
(6)
where μr is the relative recoil permeability and M is the residual
magnetization vector. By employing the scalar magnetic potential ϕ, the field behavior can be governed by a set of 2-rank
partial differential equations (Laplace/Poisson equations)
in Region I: ∇2 ϕI (r, θ) =
div M
(7)
μr
Fig. 3.
PM magnetization distribution.
decomposition over [−π, π], it yields
∞
cos (mp1 (θ0 + ωt t)) sin mp1 θ
Mr =
Zm
(9)
− sin (mp1 (θ0 + ωt t)) cos mp1 θ
m =1
Zm =
2 [1 − (−1)m ] Br
mπμ0
in Region II:
(10)
∇2 ϕII (r, θ) = 0
(11)
in Region III:
∇ ϕ
(r, θ) = 0
(12)
in Region IV:
∇2 ϕIV (r, θ) = 0.
(13)
2
III
Then, the corresponding boundary conditions can be defined as
following.
For the ith opening Oi , i = 1, 2, ..., 6
L
θiL = βi − f /2 and r ∈ [r1 , r2 ] : ϕIII
r, θi = 0 (14)
i
r, θiR = 0 (15)
θiR = βi + f /2 and r ∈ [r1 , r2 ] : ϕIII
i
r = r1 and θ ∈ [θiL , θiR ] : ϕIII
i (r1 , θ) = 0.
(16)
For the jth slot Sj , j = 1, 2
L
θjL = γj − g/2 and r ∈ [0, r2 ] : ϕIV
r, θj = 0
j
r, θjR = 0
θjR = γj + g/2 and r ∈ [0, r2 ] : ϕIV
j
r = 0 and θ ∈ [θjL , θjR ] : ϕIV
j (0, θ) = 0.
(17)
(18)
(19)
M = Mr r + M θ θ
(8)
where Mθ = 0 and Mr = Mr (θ0 , ωt , t) indicates the PM magnetization distribution shown in Fig. 3. By using Fourier series
For the air gap
r = r2 and θ ∈ [αk − e/2, αk + e/2], k = 1, 2, ..., 8 :
ϕII (r2 , θ) = 0
(20)
JIAN et al.: ANALYTICAL METHOD FOR MAGNETIC FIELD CALCULATION IN A LOW-SPEED PERMANENT-MAGNET HARMONIC MACHINE
865
r = r2 and θ ∈ [βi − f /2, βi + f /2], i = 1, 2, ..., 6:
ϕII (r2 , θ) = ϕIII
i (r2 , θ)
∂ϕII (r, θ) ∂ϕIII
i (r, θ) =
∂r
∂r
r =r 2
r =r 2
(21)
(22)
r = r2 and θ ∈ [γj − g/2, γj + g/2], j = 1, 2:
ϕII (r2 , θ) = ϕIV
j (r2 , θ)
∂ϕIV
∂ϕII (r, θ) j (r, θ) =
∂r
∂r
r =r 2
(23)
Fig. 4.
.
Region of openings.
(24)
general solution is given by
∞ (An rn p 2 + Bn r−n p 2 ) cos np2 θ
I
ϕ1 =
+ (Cn rn p 2 + Dn r−n p 2 ) sin np2 θ
r =r 2
For the PMs
r = r3 and θ ∈ [−π/p2 , π/p2 ]:
n =1
ϕI (r3 , θ) = ϕII (r3 , θ)
I II ∂ϕ ∂ϕ
μr
− Mr = μr
∂r r =r 3
∂r r =r 3
(25)
(26)
r = r4 and θ ∈ [−π/p2 , π/p2 ]:
ϕI (r4 , θ) = 0.
(27)
IV. FIELD SOLUTION
A. Field Distribution in Air Gap
The scalar potential in Region II is governed by the Laplace
equation given in (11). By separating the variables r and θ, the
general solution can be expressed as
∞ (En rn p 2 + Fn r−n p 2 ) cos np2 θ
ϕ =
+ (Gn rn p 2 + Hn r−n p 2 ) sin np2 θ
II
n =1
+ E0 ln r + G0
(28)
where Dn , En , Fn , Gn , D0 , and E0 are the Fourier coefficients
to be determined.
It indicates that the field solutions can be expressed as the sum
of several harmonics components around the circumference, and
the magnitude of each harmonic is a function of the radial length.
It should be noted that the zero harmonic term has to be taken
into account. Due to the existence of the FMPs, the magnetic
path is no longer even. Thus, the circumferential integration of
the scalar magnetic potentials should not be zero. Moreover,
considering the periodicity, the zero harmonic term should not
be related to θ.
B. Field Distribution in Permanent Magnets
The scalar potential in Region I is governed by the Poisson
equation given in (7). According to the superposition law, its
general solution is the sum of the solution of the corresponding
Laplace equation and one special solution of its own.
First, the general solution of the corresponding Laplace equation is considered. In order to satisfy the boundary conditions
given in (25), by using the method of separating variables, the
+ A0 ln r + C0 .
(29)
Second, the special solution of the Poisson equation is taken
into account. The residual magnetization given by (9) and (10)
can be rewritten as
∞
cos (np2 (θ0 + ωr t)) sin np2 θ
Mr n
Mr =
(30)
− sin (np2 (θ0 + ωr t)) cos np2 θ
n =1
⎧
⎨ 4Br , when n = iG , i = 1, 3, 5, ...
r
(31)
Mr n = (iπμ0 )
⎩
0,
otherwise.
From (7) and (8), it yields
div M
∂ (Mr ) ∂ (Mθ )
Mr
Mr
+
+
=
.
=
μr
μr r
μr ∂r
μr r∂θ
μr r
(32)
Thus, from (30) to (32), it is easy to find a special solution of
(7) given by
∞
cos (np2 (θ0 + ωr t)) sin np2 θ
I
ϕ2 =
Wn r
(33)
− sin (np2 (θ0 + ωr t)) cos np2 θ
n =1
⎧
4Br
⎨
, when n=iGr , i=1, 3, 5, . . .
Wn = iπμ0 μr [1 − (np2 )2 ]
⎩
0,
otherwise.
(34)
Finally, the general solution of the scalar potential in Region I
can be expressed as
ϕI = ϕI1 + ϕI2
⎤
⎡
An rn p 2 + Bn r−n p 2
θ
cos
np
∞
2
⎥
⎢
−Wn r sin (np2 (θ0 + ωr t))
⎥
⎢
=
⎦
⎣
np2
−n p 2
C
r
+
D
r
n
n
n =1
+
sin np2 θ
+Wn r cos (np2 (θ0 + ωr t))
+ A0 ln r + C0
(35)
where An , Bn , Cn , Dn , A0 , and C0 are the Fourier coefficients
to be determined.
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 3, SEPTEMBER 2011
Fig. 5.
Scalar magnetic potential distribution on surface of FMPs. From top to bottom: initial phase angle θ 0 = 0 and θ 0 = π/42.
Fig. 6.
Magnetic flux density in middle of air gap. From top to bottom: radial component and tangential component.
C. Field Distribution in Opening Oi
expressed as
Fig. 4 shows the detailed scheme of the region of opening.
The scalar potential in the ith opening Oi is governed by the
Laplace equations given in (12). Considering the boundary conditions listed in (17) and (18), its general solution should be
ϕIII
i =
∞ m =1
Iim rλm f +Jim r−λm f
sin λm f
θ − βi +
f
2
(36)
JIAN et al.: ANALYTICAL METHOD FOR MAGNETIC FIELD CALCULATION IN A LOW-SPEED PERMANENT-MAGNET HARMONIC MACHINE
Cn r3n p 2 − Dn r3−n p 2 − Gn r3n p 2 + Hn r3−n p 2
Mr n
r3
=
− Wn
cos (np2 (θ0 + ωr t)) .
μr
np2
867
(47)
Fourth, on the surface of the FMPs (r = r2 ), from (20),
(21), and (23), the scalar potential on this surface can also be
expressed as
Fig. 7.
Flux line distribution.
where λm f = mπ/f , and Iim and Jim are the Fourier coefficients to be determined.
D. Field Distribution in Slot Sj
The scalar potential in the jth slot Sj is governed by the
Laplace equations given in (13). Similar to the case of openings,
it is easy to know that the general solution could be expressed as
∞ g Kj m rλm g sin λm g θ − γj +
(37)
ϕIV
j =
2
m =1
where λm g = mπ/g, and Kj m is the Fourier coefficients to be
determined. The reason why only the positive exponential term
is employed lies in the assumption on the infinite depth of slots.
⎧ IV
ϕ1 (r2 , θ) ,
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 0,
ϕII (r2 , θ) = ϕIII
i (r2 , θ) ,
⎪
⎪
⎪ ϕIV (r2 , θ) ,
⎪
2
⎪
⎪
⎩ IV
ϕ1 (r2 , θ) ,
λm f
Iim r1
+
−λ
Jim r1 m f
= 0.
αi − e/2 ≤ θ < αi + e/2
βi − f /2 ≤ θ < βi + f /2
−g/2 ≤ θ < g/2
π/p2 − g/2 ≤ θ < π/p2 .
(48)
Expanding (48) into Fourier series over [−π/p2 , π/p2 ], one
gets
∞
E0
+
[En cos np2 θ + Gn sin np2 θ] (49)
2
n =1
ϕII (r2 , θ) =
p2
E0 =
π
π /p 2
−π /p 2
ϕI I (r2 , θ)dθ
⎡
⎞⎤
⎛
2
6
λm f λm g K
+f
r
I
gr
j
m
im
2
2
⎢1−(−1) ⎜
⎟⎥
p2
j =1
i=1
⎢
⎟⎥
⎜
=
⎣
⎠⎦
⎝
6
π m =1
mπ
−λm f +f r2
Jim
∞
E. Boundary Conditions
In what follows, the boundary conditions are discussed so as
to determine the parameters defined in the solutions.
First, at the bottom surface of the ith opening Oi (r = r1 ),
from (16) and (36), it yields
−π/p2 ≤ θ < −π/p2 + g/2
m
i=1
En =
p2
π
(50)
π /p 2
−π /p 2
ϕI I (r2 , θ) cos np2 θ dθ
⎤
−λm f
λm f
τ
(m,
n,
f,
i,
β)
I
r
+J
r
im
im
2
2
⎥
p2 ⎢
⎥
⎢ i=1 m =1
=
2
∞
⎦
⎣
π
λ
+
τ (n, m, g, j, γ)Kj m r2 m g
(38)
Second, on the outside surface of the PMs (r = r4 ), from (27)
and (35), it yields
⎡
6 ∞
j =1 m =1
A0 ln r4 + C0 = 0
(39)
An r4n p 2 + Bn r4−n p 2 − Wn r4 sin (np2 (θ0 + ωr t)) = 0
(40)
Cn r4n p 2
+
Dn r4−n p 2
+ Wn r4 cos (np2 (θ0 + ωr t)) = 0. (41)
Third, on the inside surface of the PMs (r = r3 ), from (25), (26),
(28), and (35), it yields
A0 ln r3 + C0 = E0 ln r3 + G0
An r3n p 2
+
Bn r3−n p 2
(42)
= Gn r3n p 2 + Hn r3−n p 2
=
−π /p 2
ϕI I (r2 , θ) sin np2 θ dθ
⎤
−λm f
λm f
σ(m,
n,
f,
i,
β)
I
r
+J
r
im
im
2
2
⎥
p2 ⎢
⎥
⎢ i=1 m =1
=
2
∞
⎦
⎣
π
λm g
σ(n, m, g, j, γ)Kj m r2
+
⎡
6 ∞
(52)
(44)
A0 = E0
−
π /p 2
(43)
Cn r3n p 2 + Dn r3−n p 2 + Wn r3 cos (np2 (θ0 + ωr t))
An r3n p 2
p2
π
(51)
j =1 m =1
− Wn r3 sin (np2 (θ0 + ωr t))
= En r3n p 2 + Fn r3−n p 2
Gn =
where τ (·) and σ(·) are defined by (53) and (54), as shown at
the bottom of the next page.
Thus, from (28) and (49), it yields
(45)
Bn r3−n p 2
Mr n
Wn −
μr
−
En r3n p 2
+
r3
sin (np2 (θ0 + ωr t))
np2
E0 lnr2 + G0 = E0
Fn r3−n p 2
En r2n p 2
(46)
+
Fn r2−n p 2
(55)
= En
Gn r2n p 2 + Hn r2−n p 2 = Gn .
(56)
(57)
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Fig. 8.
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 3, SEPTEMBER 2011
Asynchronous harmonic components of radial flux density in middle of air gap.
TABLE I
ASYNCHRONOUS HARMONICS
Similarly, from (22), (24), and (49), one gets
∞
E0 (1 − (−1)m ) +
np2
λm f
n =1
τ (m, n, f, i, β) En r2n p 2 − Fn r2−n p 2
×
+σ (m, n, f, i, β) Gn r2n p 2 − Hn r2−n p 2
mπ −λ
λ
(58)
Iim r2 m f − Jim r2 m f
=
2
∞
E0 (1 − (−1)m ) +
np2
λm g
n =1
τ (m, n, g, j, γ) En r2n p 2 − Fn r2−n p 2
×
+σ (m, n, g, j, γ) Gn r2n p 2 − Hn r2−n p 2
=
mπ
λ
Kj m r2 m g .
2
(59)
Consequently, all the Fourier coefficients can be determined
by (38)–(47) and (55)–(59). After the scalar potentials in each
region are obtained, the flux density can be deduced by
∂ϕ
Br = −μ0
∂r
1 ∂ϕ
Bθ = −μ0
r ∂θ
(60)
(61)
where Br and Bθ represent the radial and tangential components
of the flux density, respectively.
V. CALCULATION RESULTS
Fig. 5 shows the scalar potential distribution along the surface
of the FMPs (r = r2 ) with different initial phase angle θ0 . The
first harmonic component (PPN = 3) and the seventh harmonic
component (PPN = 21) are also shown. It can be found that when
the seventh harmonic component moves rightward by π/42 with
the rotor, the first harmonic component will move leftward by
π/6. This agrees well with the magnetic-gearing effect given
in (4).
⎧
u
u m
λ
cos
np
Γ
Γ
−
(−1)
−
cos
np
+
⎪
m
u
2
v
2
v
⎪
⎪
2
2 ,
⎪
(λm u = np2 )
⎨
2
2
λ
−
(np
)
2
m
u
τ (m, n, u, v, Γ) =
u
u
mπ
⎪
⎪
− (−1)m cos np2 Γv +
− λm u Γv
u sin
cos np2 Γv −
⎪
⎪
2
2 +
2
⎩
, (λm u = np2 )
2 (λm u + np2 )
2
⎧
u
u ⎪
λm u sin np2 Γv −
− (−1)m sin np2 Γv +
⎪
⎪
2
2 ,
⎪
(λm u = np2 )
⎨
2
2
λ
−
(np
)
2
m
u
σ (m, n, u, v, Γ) =
mπ
u
u
⎪
m
⎪
−
λ
Γ
u
cos
−
sin
np
+
sin
np
Γ
Γ
−
(−1)
mu v
⎪
2
v
2
v
⎪
2
2
2 +
⎩
, (λm u = np2 )
2 (λm u +n p 2 )
2
(53)
(54)
JIAN et al.: ANALYTICAL METHOD FOR MAGNETIC FIELD CALCULATION IN A LOW-SPEED PERMANENT-MAGNET HARMONIC MACHINE
Fig. 6 shows the magnetic field density in the middle of the
air gap. In order to assess the validity of the proposed analytical method, the corresponding results derived from the FEM
are provided for comparison. When using the FEM, the calculation zone is divided into 11 320 triangular elements, and then the
Newton–Raphson (N–R) method combined with the incomplete
Cholesky-conjugate gradient (ICCG) algorithm is used to solve
the nonlinear equations. In order to test the infinite permeability
assumption, two cases have been taken into consideration: In
case 1, the infinite permeability assumption is adopted for the
iron yokes. In case 2, the B-H characteristic of the laminated
silicon steel (Type 50H470) is engaged for the iron yokes. The
calculated results did not show significant difference between
these two cases, which demonstrates that the infinite permeability assumption is acceptable in this model. Fig. 7 illustrates
the flux line distribution obtained by using the FEM. It can
be seen that the analytical results closely agree with the FEM
results.
Fig. 8 shows the dominant asynchronous harmonic components of the radial flux density in the middle of the air gap. Their
rotational magnitudes and the resulted gear ratios are listed
in Table I. It can be found that the first harmonic component
(PPN(1,−1) = 3) is of high strength and could result in remarkable speed variation. More importantly, it has the same PPN
with the fundamental component of the electromagnetic field
excited by the armature windings; thus, stable torque transmission and energy conversion can be achieved through their
interaction.
VI. CONCLUSION
In this paper, an analytical approach to calculate the magnetic field distribution in a low-speed PM harmonic machine
is developed. First, the magnetic-gearing effect from which the
harmonic machine is derived is discussed. Second, a seriesslot model which is composed of a group of partial differential equations concerning the scalar magnetic potential is built
up. Third, the field solutions are obtained by using the method
of separating variables and analyzing the field boundary conditions. Finally, the flux densities are derived from the scalar
potentials. All the results agree well with those obtained from
the FEM.
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870
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 3, SEPTEMBER 2011
Linni Jian (S’07–M’09) received the B.Eng. degree
from the Huazhong University of Science and Technology, Wuhan, China, in 2003, the M.Eng. degree
from the Institute of Electrical Engineering, Chinese
Academy of Sciences, Beijing, China, in 2006, and
the Ph.D. degree from The University of Hong Kong,
Pokfulam, Hong Kong, in 2010.
He is currently an Assistant Professor at Shenzhen Institutes of Advanced Technology, Chinese
Academy of Sciences. His research interests include
the areas of electric drives, electric vehicles, power
electronics, and smart grid. In these areas, he has published more than 20 referred technical papers.
Guoqing Xu received the B.Sc., M.Sc., and Ph.D.
degrees in electrical engineering from Zhejiang University, Hangzhou, China, in 1988, 1991, and 1994,
respectively.
He was a Postdoctoral Researcher with Northern Jiaotong University, Beijing, China. He was with
Tongji University, Shanghai, China, where he joined
in 1997 as an Associate Professor and Professor in the
Department of Electrical Engineering, and was promoted as the Chair of the Department of Electrical
Engineering until 2003. From 2006 to 2007, he was
a Visiting Professor in the Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong (CUHK), Hong Kong,. Since
2007, he has been a Research Professor with the Mechanical and Automation
Engineering, CUHK. He has been an Associate Director of CUHK-CAS Shenzhen Institutes of Advanced Integration Technology, CUHK, since 2007. He has
been the Chief Scientist of R&D Centre for Electric Vehicle, Chinese Academy
of Sciences, China, since 2009. He has also been the Director of CUHK-CAS
Shenzhen Institutes of Advanced Integration Technology, CUHK, since 2010.
He has published more than 100 referred papers in international journals and
conferences. His current research interests include electric vehicle control, energy processing, and automotive electronics.
Chunting Chris Mi (S’00–A’01–M’01–SM’03)
received the B.S.E.E. and M.S.E.E. degrees
from Northwestern Polytechnical University, Xi’an,
China, and the Ph.D. degree from the University of
Toronto, Toronto, Ontario, Canada, all in electrical
engineering.
He is currently an Associate Professor Electrical and Computer Engineering, and the Director of
DTE Power Electronics Laboratory at the University
of Michigan-Dearborn, Dearborn. Previously, he was
an Electrical Engineer with General Electric Canada,
Inc. He has conducted extensive research and published more than 100 articles.
His research interests include electric drives, power electronics, electric machines, renewable energy systems, and electrical and hybrid vehicles.
Dr. Mi is the recipient of the National Innovation Award, Government Special
Allowance Award, Distinguished Teaching Award, and Distinguished Research
Award of the University of Michigan-Dearborn. He is also a recipient of the
2007 IEEE Region 4 Outstanding Engineer Award, the IEEE Southeastern
Michigan Section Outstanding Professional Award, and the SAE Environmental Excellence in Transportation (E2 T) Award. He is the Chair of the IEEE
Southeastern Michigan Section (2008–2009). He was Vice Chair of the IEEE
Southeastern Michigan Section from 2006 to 2007, and Chair in 2008. He was
the General Chair of the fifth IEEE Vehicle Power and Propulsion Conference held in Dearborn, in September 6–11, 2009. He is an Associate Editor
of the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, an Associate Editor of the IEEE TRANSACTIONS ON POWER ELECTRONICS-–LETTERS, a Senior
Editor of the IEEE VEHICULAR TECHNOLOGY MAGAZINE, a Guest Editor of
the International Journal of Power Electronics, an Editorial Board Member of
the International Journal of Electric and Hybrid Vehicles and IET Electrical
Systems in Transportation, and an Associate Editor of the Journal of Circuits,
Systems, and Computers (2007–2009).
K. T. Chau (M’89–SM’04) received the B.Sc.(Eng.)
(first-class Hons.), M.Phil., and Ph.D. degrees in electrical and electronic engineering from The University
of Hong Kong, Pokfulam, Hong Kong, in 1988, 1991,
and 1993, respectively.
He is currently a Professor in the Department of
Electrical and Electronic Engineering, and the Director of International Research Center for Electric
Vehicles, The University of Hong Kong. His teaching and research interests focus on three main areas:
electric vehicles, electric drives, and power electronics. In these areas, he has published more than 200 refereed technical papers.
He is also a coauthor of a monograph, Modern Electric Vehicle Technology
(London, U.K.: Oxford Univ. Press, 2001).
Prof. Chau is a Fellow of the IET. He was the recipients of the Outstanding
Young Researcher Award in 2003, the University Teaching Fellowship Award
in 2004, and the Award for Innovative Excellence in Teaching, Learning and
Technology in 2005.
C. C. Chan (M’77–SM’77–F’92) received the B.Sc.,
M.Sc., and Ph.D. degrees in electrical engineering
from China University of Mining & Technology,
Tsinghua University and The University of Hong
Kong, in 1957, 1959, and 1982, respectively.
He is currently the Honorary Professor and the former Head of the Department of Electrical and Electronic Engineering, The University of Hong Kong,
Pokfulam, Hong Kong. He has had more than ten
years industrial experience and more than 35 years
academic experience.
He is the Founding President of the International Academy for Advanced
Study, China, the Cofounder and Rotating President of the World Electric Vehicle Association, and the President of the Electric Vehicles Association of Asia
Pacific. He is a Fellow of the Royal Academy of Engineering, U.K., the Chinese
Academy of Engineering, The Ukraine Academy of Engineering Sciences, IEE
and HKIE.